Bioequivalence and Bioavailability Forum

Hi David & Darya,

❝ For some reason, I can't see the image with the formula that you put on the first post.

I uploaded a copy. Should be visible by now.

❝ ❝ The only thing about what I'm not sure is CV. Because in formula that I used this is intra-subject variability, but in PowerTOST() this is coefficient of variation as ratio. In calculations in both cases I used СV - 0.3.

❝

❝ Within subject standard deviation and within subject CV are different parameters. Nevertheless, I think that this is not the only reason for such a big difference.

Chow used the standard deviation, whilst in Power.TOST the CV (fraction, not in %!) is used. However, no big difference since \(CV_w = \sqrt{e^{s_{w}^{2}} - 1}\) and the other way ’round \(s_w = \sqrt{\log{(CV_{w}^{2}) + 1}}\). In Power.TOST for convenience you can use se2CV(foo) for the former and CV2se(foo) for the latter.

❝ There is a sentence in one of the articles that is quoted on SampleNTOST formula that may clarify this issue: […]

❝ So by reading this, I am not sure if Chow formula might be appropriate to calculate sample size for BABE trials.

I think it’s crap. The formula Darya posted (10.2.6) of p.259 of the book gives the impression that n is the total sample size. The text continues with:

Since the above equations do not have an explicit solution, for convenience, for a 2 × 2 crossover design, the total sample size needed to achieve a power of 80% or 90% at 5% level of significance with various combinations of ε and δ is given in Table 10.2.1.

However, in the example which follows on p.260 we read:

By referring to Table 10.2.1, a total of 24 subjects per sequence is needed in order to achieve an 80% power at the 5% level of significance.

(my emphases)

❝ […] Perhaps dlabes might clarify this, since he is the master that we all should thank for the amazing PowerTOST package

Detlew is on vacation. Some clarifications:

Zhang¹
(5) where n = sample size per sequence

(9) with a correction term c


Hauschke et al.²


Chow & Liu (5.4.10)³

Using the formulas of Zhang or Chow & Liu, you get the sample size / sequence. To obtain the total, multiply by 2, which is already done it the right-hand side of Hauschke’s formulas.

Comparison with the references:
Table 2, untransformed data, 90% Power, Δ 0.2, σ 0.3, ∕θ 0.05 (p.537¹): 37 / sequence

library(PowerTOST)
sampleN.TOST(CV=se2CV(0.3), theta0=0.05, theta1=-0.2, theta2=0.2,
             logscale=FALSE, targetpower=0.90,
             print=FALSE)[["Sample size"]]/2
[1] 37

Table 5.4.1, untransformed data, 80% Power, Δ 0.2μ_R, α 0.05 (p.158³): 52

library(PowerTOST)
sampleN.TOST(CV=0.30, theta0=0.05, theta1=-0.2, theta2=0.2,
             logscale=FALSE, targetpower=0.80,
             print=FALSE)[["Sample size"]]
[1] 52

Table 5.1, log-transformed data, 80% Power, (θ₁, 1∕θ₁) = (0.80, 1.25), θ 0.95, α 0.05 (p.113²): 40

library(PowerTOST)
sampleN.TOST(CV=0.30, theta0=0.95, theta1=0.8, theta2=1.25,
             targetpower=0.80,
             print=FALSE)[["Sample size"]]
[1] 40

Stop using the formula given by Chow, Shao, Wang! Sample sizes are way too small – which compromises power. If you used it in the past for 80% power – if all assumptions (CV, θ₀) turned out to be “correct” – substantially more than 20% of studies should have failed. If not, you were lucky!

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1. Zhang P. A Simple Formula for Sample Size Calculation in Equivalence Studies. J Biopharm Stat. 2003;13(3):529-38. doi:10.1081/BIP-120022772
   
2. Hauschke D, Steinijans V, Pigeot I. Bioequivalence Studies in Drug Development. Chichester: John Wiley; 2007. p.116. doi:10.1002/9780470094778
   
3. Chow S-C Liu J-p. Design and Analysis of Bioavailability and Bioequivalence Studies.
   Boca Raton: Chapman & Hall/CRC Press; 3^rd ed. 2009. p.157. doi:10.1201/9781420011678